John Trangenstein: Math 225 Scientific Computing II

HyperText Scientific Computing Class Notes (Local users only)

John Trangenstein : Math 225 Scientific Computing

Office hours: MWF 9:00-10:00 Physics 029C
Class hours: MWF 3:00-4:15 Physics 219 Syllabus:

Interpolation and Approximation
- Polynomial Interpolation
- Splines
- Least Squares Approximation
Differentiation and Integration
- Numerical Differentiation
- Numerical Integration
- Gaussian Quadrature
- Adaptive Quadrature
Initial Value Problems
- Linear Equations with Constant Coefficients
- Linear Multistep Methods
- Runge-Kutta Methods
- Extrapolation Methods
- Stiffness
Boundary Value Problems
- Shooting Methods
- Finite Difference Methods
- Finite Element Methods

News flash(es):

To enable your sessions to dump cores in bash, add
ulimit -Sc unlimited
to your .cshrc file.
To print an xmgr plot, type (in another window)
xwd -nobdrs | xpr -device ps | lpr
You will have to click in the xmgr window to choose what gets printed.

Course grade depends on weekly homework assignments
All homework must contain

a written description of the problem and numerical approach
a copy of the code
pictures or numbers for numerical results, and
a discussion of the results (e.g., describe the accuracy, convergence rate, ...)

To get online help with commands and utilities, there are several useful sources:

Gnu documentation. Type "info". When the window comes up, scroll to your favorite topic (e.g., "Make" or "g77"). Use the middle mouse button on the left border to scroll and click.
Online books.
Netlib Home Directory

Also, the Basic Linear Algebra Subroutines (BLAS) are available in ~johnt/math224/fblas, and the Fortran versions of the LaPack routines are available in ~johnt/math224/flapack.
How to set up an animated web page:

generate a .gif file containing the images; this could be done using the convert command; for example,
convert *.xpm movie.gif
make sure that the movie.gif file is readable by everyone, and that all of the directories containing this file are readable and executable by everyone.
make sure that you have a directory named "public_html" in your home directory; if not, type
cd
mkdir public_html
Also make sure that this directory is accessible by everyone:
chmod a+rx public_html
You can copy this web page to your home directory to get started:
cp ~johnt/public_html/math224.html ~/public_html/math225.html
Make sure that everyone can read this web page:
cd public_html
chmod a+r math225.html
You can probably guess at the meanings of the html commands. For example, ul begins an unordered list, and /ul ends it. The commands are contained inside less than and greater than signs. The command to display the movie looks like this: $ANIMATION NOT VIEWABLE OUTSIDE DUKE$

Homework (unless stated otherwise, all assignments can be found in the online HyperText Scientific Computing Class Notes ):

Due January 22:
- Section 9.2 #3
- VR Homework Results
- DS Homework Results
- MB Homework Results
- MJ Homework Results
- DC Homework Results
Due January 29:
- Section 9.3.5 #1
  Write the Hermite cubic interpolant routine in a separate file from the main program that sets up the arrays and writes the results. Also write a Makefile that compiles the files and generates the executable.
- VR Homework Results
- MB Homework Results
- MJ Homework Results
- DC Homework Results
- YL Homework Results
Due February 5:
- Section 9.4 #1
- Section 9.4 #2
- DS Homework Results
- VR Homework Results
- MB Homework Results
- DC Homework Results
- YL Homework Results
Due February 12:
- Write a routine to compute the zeros of the Legendre polynomials of arbitrary order.
- DS Homework Results
- VR Homework Results
- YL Homework Results
- DC Homework Results
- MB Homework Results
Due February 19:
- Section 11.2 #3
- YL Homework Results
- DS Homework Results
- VR Homework Results
- MJ Homework Results
- MB Homework Results
- DC Homework Results
Due March 5: Program epsode to solve the following problems:
- the two-body problem
  x''=-x/r^3, y''=-y/r^3 where r=sqrt(x^2+y^2)
  x(0)=1-epsilon, y(0)=0
  x'(0)=0, y'(0)=sqrt((1+epsilon)/(1-epsilon))
  where epsilon=0.9 is the eccentricity of the orbit
  the time of interest is t=20
- the forced electrical circuit with nonlinear inductor
  Q'=I, I'=-I/10 - Q^3 + b cos(t)
  Q(0) = 3.3, I(0)=-3.3
  where b=9.8 leads to a periodic orbit and b greater than 10 leads to chaotic behavior
  times of interest are t=2 * pi * k, where k less than 20 is an integer
Run your code with error tolerance close to machine precision to obtain the "exact" solution for each problem. Then run your code with tolerances given by 10^n where n=-1,-2,.... Plot the log of the error versus the log of the computational time. Comment on the results. In particular, how good are the various error tests (absolute error, relative error in current solution, relative error in max over time) in controlling the actual error? Can you get any information about how epsode varies order and stepsize internally while solving these problems?
Due March 19:
- Program finite differences to solve
  -y'' = (pi/2)^2 sin(pi*x/2) , x in (0,1)
  y(0) = 0 , y'(1) = 1
  Use Lapack routine dgttrs to solve the linear system. You do not need to compile dgttrs and dgttrf; it is sufficient to link with "-llapack".
- Use your code to perform a mesh refinement study. Plot the log of the mean square error at the mesh points versus minus the log of the mesh width. Also plot the log of the error in the solution at x=1 versus minus the log of the mesh width. Finally, plot the log of the error in the derivative of the solution at x=0 versus minus the log of the mesh width. How does the observed error in the numerical results compare with the order you should expect to observe?
- Due March 26:
  Section 12.5.3 #2
- Due April 9:
  Section 12.5.5 #3
Return to: John A. Trangenstein *
Last modified: